py

Modeling

Overview

Data fitting

Fig. 1 illustrates the obtained fits, where we have denoted the biobricks by their last four letters for convenience. It can be seen that all growth curves resemble a sigmoid and that the Gompertz function managed to properly capture the structure. Furthermore, it can be noted that there is a small spread of outliers on some measurements, most notably at around 3 hours in the strain carrying 2011.

Figure 1

Figure 1: The growth curves along with the fitted Gompertz function.

Bootstrap

Proceeding with the bootstrap, the residuals were first calculated and subjected to the quantile filter. Only the growth curves corresponding to 2011, 2015 and 2016 contained outliers and at most, two samples were removed. The results are illustrated in fig. 2. The dark blue central curves in the plots to the left are the estimates obtained from the previous Gompertz fit and the light blue regions correspond to the bootstrapped sigmoids plotted on top of each other. These can be interpreted as new realizations obtained from the residuals and the intervals they span correspond to the spread of the samples. The figures to the right illustrate the histograms of the residuals and it can be seen that they are all of approximately the same magnitude and centered around zero with possibly a slight skewness. Furthermore, by visually inspecting fig. 1, it can be seen that the magnitude and change of sign of the residuals are approximately stationary over the entire growth curves. This justifies our approach bootstrapping to estimate the coming growth parameters.

Figure 2

Figure 2: The bootstrap samples of the growth growth curves in fig. 1. The dark blue sigmoid corresponds to the estimated growth curve and the light blue corresponds to the bootstrap samples.

The histograms for the bootstrapped growth parameters are illustrated in fig. 3. It can be seen that the distributions for all parameters and experiments are well-behaved with a clear central mass and a corresponding surrounding and decreasing mass. This indicates that the Gompertz sigmoid is stable with respect to this problem and is therefore a good model of choice when also considering the residuals and fits in fig. 2. Furthermore, it is worth noting that all parameter distributions are not gaussian due to the skewness.

Figure 3

Figure 3: The bootstrapped parameters.

Parameter estimates

The final parameter estimates along with their 95% and 90% two-sided confidence intervals are shown in table 1, fig. 4 and fig. 5 respectively. By looking at fig. 4, it can be seen that there is no significant difference in the stationary cell densities between the samples used. This may be due to the small cultivation volumes as well as the large headspaces used in the experiment which causes a high mass transport of oxygen. In this way, oxygen might dissolve sufficiently fast and consequently mask the effect of VHb.

Looking at the time to exponential growth, we see that the confidence intervals cover both negative and positive times for all samples except 2013 and 2016, indicating that for all the other samples, we cannot be certain that we started measuring during the lag phase despite the low initial OD. Nevertheless, this does not change the results for the other parameters since the growth rate is given by the slope, which is constant throughout the measurement period, and the stationary cell density is independent of when the measurements started. As also seen in fig 5., the variance is too high to draw any conclusions.

As for the growth rate during the exponential phase, all VHb-expressing strains had a higher average growth rate than the negative control, but only 2014 and 2015 had significantly higher growth rates at a confidence level of 95%. Interestingly these two have the lowest promoter strengths, with the relative strengths 0.00 and 0.01, respectively. However, we have results suggesting that the expression level in 2014 is not 0, but that the promoter is in fact leaking to some extent. The implication of this is that a low level of VHb expression has a positive effect on the growth rate. While the averages for the higher-expression strains are also higher than for the negative control, the difference is not significant on a 95% confidence level and conclusions about their effect cannot be drawn.

Table 1: The estimated parameters along with their confidence intervals.
Sample	Stationary OD	95% CI	Growth rate	95% CI	Lag time	95% CI
2010	6.12	(5.87, 6.67)	1.71	(1.51, 2.03)	0.42	(-0.26, 0.79)
2011	6.65	(6.36, 7.65)	1.59	(1.39, 1.95)	0.50	(-0.56, 1.04)
2013	6.04	(5.83, 6.36)	1.66	(1.47, 1.89)	0.51	(0.08, 0.83)
2014	6.45	(6.10, 7.50)	1.83	(1.61, 2.18)	0.24	(-0.65, 0.70)
2015	6.42	(6.18, 7.06)	1.78	(1.62, 1.95)	0.23	(-0.31, 0.50)
2016	6.52	(6.22, 6.80)	1.68	(1.50, 1.90)	0.51	(0.14, 0.86)
R0011	6.08	(5.79, 6.46)	1.28	(1.15, 1.47)	0.55	(0.03, 0.97)

Figure 4

Figure 4: The estimated parameters along with their 95 % confidence intervals.

Figure 5

Figure 5: The estimated parameters along with their 90 % confidence intervals.

Finally, fig. 6 shows a scatter plot of the estimated growth rates for each biobrick with respect to the promoter strengths. It can be seen that there is a slight negative correlation which shows that while VHb increases the growth rate at all expression levels, the increase becomes less prominent as the expression level increases. This may be due to the additional taxation put on the cell economy during the initial growth phase outweighing the positive effect of VHb and consequently slowing the cell growth. However, while the results may be explained from a mechanistic point of view, they are not entirely consistent with previous research. It has been shown that an increased level of VHb consistently increases the growth rate in E. coli when modulating the expression level through an IPTG-inducible lac-regulated promoter [1]. However, the results were based on microaerobic conditions. On the other hand, as our cultivations were done in flasks instead of fermentors, we could not control the oxygenation to the same extent, especially during sampling events when flasks needed to be opened. Additionally, the headspace in our cultivations was 80%. This means that we most likely did not have consistently microaerobic conditions.

Figure 6: The growth rate versus the promoter strength.

Conclusion

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References

[1] Tsai PS, Hatzimanikatis V, Bailey JE. (1996). Effect of Vitreoscilla Hemoglobin Dosage on Microaerobic Escherichia coli Carbon and Energy Metabolism. Biotechnology and Bioengineering, 49, 139-150.

[1] [1] Tsai PS, Hatzimanikatis V, Bailey JE. (1996). Effect of Vitreoscilla Hemoglobin Dosage on Microaerobic Escherichia coli Carbon and Energy Metabolism. Biotechnology and Bioengineering, 49, 139-150.

Team:Lund/Model/GrowthCurves/Results